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For any two sets, prove that:

Question:

For any two sets, prove that:

(i) $A \cup(A \cap B)=A$

(ii) $A \cap(A \cup B)=A$

Solution:

(i)

$\mathrm{LHS}=A \cup(A \cap B)$

$\Rightarrow \mathrm{LHS}=(A \cup A) \cap(A \cup B)$

$\Rightarrow \mathrm{LHS}=A \cap(A \cup B) \quad(\because A \subset A \cup B)$

$\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$

(ii)

$\mathrm{LHS}=A \cap(A \cup B)$

$\Rightarrow \mathrm{LHS}=(A \cap A) \cup(A \cap B)$

$\Rightarrow \mathrm{LHS}=A \cup(A \cap B)$

$\Rightarrow \mathrm{LHS}=A=\mathrm{RHS}$

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